Increase object size/dimension across two axies, while keeping proportions
Close, but only in this case

Select these vertices/edges (you can select one group of the 8, then do Shift+G --> select similar length:



Scale along the two axes that you want in bounding box pivot mode. Then switch to individual origins and scale in the opposite direction with only these vertices selected.

Python Script

It looks like you are only wanting to move each vertex a constant distance from the center. With normal scaling from a point p, a vertex 2 meters from p will move twice as much distance as a vertex 1 meter from p. This small snippet of Python code will not do this, but rather move each vertex a constant addend, not factor, of the radius.

Note: This script scales from the object's origin. Also, you should be in object mode to run it.

import bpy
import bmesh
from math import sqrt

scale = 2  # scale variable

# prepare bmesh
mesh = bpy.context.object.data
bm = bmesh.new()
bm.from_mesh(mesh)

for v in bm.verts:
    pos = v.co.xyz

    # r = sqrt(x^2 + y^2 + z^2)
    radius = sqrt(
        pow(pos.x, 2) + 
        pow(pos.y, 2) +
        pow(pos.z, 2))

    a = (radius + scale) / radius

    # modify the two axes you want
    pos.x *= a
    pos.y *= a

    # apply new position
    v.co.xyz = pos

# make the bmesh the object's mesh
bm.to_mesh(mesh)  
bm.free()